Introduction to Practical E-M Design
The term “E-M devices” includes, but is not limited to: passive electrical devices such as transformers, inductors, and delay lines; and electromechanical devices such as motors, generators, relays, solenoids, and the “rail gun.” Some of these E-M devices also include permanent magnetic (PM) components that work synergistically with the E-M components to hold, lift, or torque magnetic susceptible material. PM components are also used to favorably change the magnetic material's magnetic saturation characteristic. Permanent magnets, PM, may also be used in magnetic devices without electro-magnetics (E-M).
All conventional E-M devices consist of a magnetizing current, IM(f), of an operating frequency (f) flowing in a conductive coil around and external to the magnetic core. The heart of all E-M devices and permanent magnetic devices is a magnetic core. The core may be made of grain oriented silicon steel, amorphous metal, ferrite, or other ferrous based materials. Some magnetic cores are a dielectric material such as plastic or air and have no ferrous enhancement of its magnetic permeability (μ) or limitations on its maximum flux density (BMx(f)).
A magnetic device determines its operational power from the steady state operating voltage (V(f)) which develops a steady state operating load current (IL(f)) through the device. The steady state power (P(f)) required by the device is the product of its operating voltage, V(f), and load current, IL(f).P(f)=V(f)*IL(f).
Magnetic devices are usually designed so the magnetizing current, IM(f), is small and negligible with respect to the load current, IL(f). The device's maximum steady safe state power capability (PMx(f)) is the product of the device's maximum safe steady state voltage (VMx(f)) and its maximum safe steady state load current (ILx(f)).PMx(f)=VMx(f)*ILx(f).
Power density, PD(f), at an operating frequency, f, is the maximum safe steady state power required by the E-M device divided by the device's magnetic material volume (vol).PD(f)=PMx(f)/vol.Maximum Current, IMx(f), and Maximum Voltage, VMx(f)
The maximum operating electrical power for all these devices is determined by either the maximum current rating of the magnetic wire forming the magnetics' coil which conducts the maximum load current, ILx(f), or the maximum operating voltage rating, VMx(f), at which the maximum flux density, BMx(r), is less than Bsat throughout all sections of the magnetic core. (BMx(r)≦Bsat) Optimal magnetics' power design, which minimizes material requirements for the magnetic core and coil, occurs when both the maximum load current, ILx(f), and the maximum voltage, VMx(f), are the device's simultaneous operating power limitations—indicates all of the coil and all of the core are efficiently used.
The magnetic core's current limitation is principally affected by the diameter of wire required for the core's coil. The product of the wire's cross sectional area (Awr) and the number (N) of required coil turns determines, to a first order, the core's minimum required window opening to accommodate the coil winding. Optimal magnetic design requires the smallest practical coil winding window opening.
All magnetic materials are characterized by their ability to accommodate the magnetic flux density, B, induced by the magnetic force field (AT, Ampere *Turn) permeating their space. This ability is known as the material's magnetic permeability (μ). A material's magnetic permeability, μ, is the product of the permeability of free space, μo, and the material's relative permeability to free space, μR. (μ=μo*μR). The permeability of free space, μo, has the value 1.26*10−6 Henries per meter (H/m), while the material's relative permeability, μR, is an integer with a range of 1 to greater than a million. Ferrous based materials designed for magnetics have a relative permeability, μR, much greater than 50—usually 1,000 to 20,000. Most magnetic materials have non-linear permeabilities increasing on the order of a factor of 10, when their magnetic force field, AT, changes from a low level magnetic excitation (ATLo) to the material's maximum high level magnetic excitation (ATHi), just below the material's maximum magnetic flux density, Bsat.
Some magnetic cores with a dielectric material such as plastic or air have no ferrous enhancement on its magnetic permeability (μ). Also, they do not have the ferrous limitation of magnetic saturation, Bsat. Air and most dielectrics have a relative permeability, μR, of approximately 1.
A magnetic material's maximum magnetic flux density, Bsat, is the maximum number of magnetic flux (φMx) lines per unit cross sectional area (AC) of magnetic material that the material will support without magnetically saturating. Magnetic force fields, AT, that try to cause the magnetic material's flux density, B, to exceed Bsat will cause the magnetic material to go into magnetic saturation and essentially reduce the magnetic core's relative magnetic permeability, μR, to 1, the value of an air core. The magnetic device's maximum operating voltage, VMx(f), occurs when the operating voltage, V(f), causes the maximum magnetizing current (IMx(f)) to induce into the magnetic device the maximum magnetic flux (φMx(f)) which causes the radially distributed magnetic flux density, BMx(r), to reach Bsat, regardless of where it occurs along the device's radial cross sectional magnetic flux density distribution, BMx(r).
In the maximum magnetic material radial cross sectional flux density distribution curves, BMx(r), shown in the Figures, the following assumptions are in place. All conventional radial flux density distribution curves, BMx(r), are normalized, maximum, and simple Amperian or the summation of normalized, maximum, simple Amperian curves. A normalized flux density distribution curve means that the actual flux density distribution, B(r), is divided by the magnetic material's magnetic saturation flux density, Bsat. Maximum flux density means that the highest flux density value of the flux density distribution, BMx(r), is Bsat and occurs in conventional magnetic material at its inner most magnetic boundary, the effective radius of the inner diameter, rIDe. Amperian may be defined as the radial cross sectional maximum flux density distribution curve, BMx(r), that follows Ampere's Law and is hyperbolically shaped, radially, from the inner boundary, rIDe, to the outer boundary, the effective radius of the outer diameter, rODe, regardless of the core's shape or size. Also, the inner boundary, rIDe, completely surrounds the magnetizing current source, IM(f), inducing the magnetic force field, AT, into the magnetic material.
On the other hand, all power density, (PD), enhanced redistributed flux density curves (BBMx(r)) presented herein, are the optimal summation of radially shifted, normalized, maximum, simple Amperian curves, BMx(r).
When an E-M or PM device's PD is compared, the device is assumed to be operating in the steady state, unless otherwise noted. A device's steady state assumes a steady electrical magnetizing current, IM(f), for a fixed load after a device has been subjected to the application of a fixed voltage, V(f), at a fixed frequency, f. Operating frequency, f, has the range of zero (0) to infinity (∞). When f equals zero (f=0), the DC or time invariant condition is being considered. Thus, VDC=V(f) when f=0.
The comparative PD of E-M and PM devices in the transient state, occurs when the device's magnetizing current, IM(t), in the time domain (t) electrically responds to a voltage step function excitation, V(t). The transient state voltage, V(t), of an electro-magnetic device is defined over its actuation time, beginning at start, t=0, to finish time, t=TD.
A circular toroid will be used to generally define the inner and outer boundaries for the radial magnetic operating regions of all magnetic cores—the region is defined from the effective radius of the inner diameter, rIDe, to the effective radius of the outer diameter, rODe The circular toroidal shape's uniform structure lends itself to easy mathematical analysis (using Ampere's Law) from which all magnetic flux distribution curves, herein, have been ideally determined. All maximum normalized flux density distribution curves, BMx(r), represent maximum operational flux density, BMx(f), at operational frequency (f). Whether the operating voltage is steady state, V(f), or transient, V(t), the flux density distribution is Amperian. The square core's magnetic flux distribution is a summation of bi-lateral Amperian cross sectional magnetic flux distributions, each being derived from an equivalent circular toroidal shape with the same inductance and material volume of the square core.
A circular toroid exhibits a precise, circular, magnetic core geometry, and as such, the magnetic flux's center for its effective radius of curvature is exactly the geometric center of the toroid. The geometry of a circular toroidal magnetic core precisely lines up with the natural circular geometry of magnetic flux lines generated by the magnetizing current, IM(f), flowing through the center of the toroid. Consequently, the effective radius of the inside diameter, rIDe, equals the geometric radius of the inside diameter (rID). Likewise, the effective radius of the toroid's outer diameter, rODe, equals the geometric radius of its outside diameter, rOD. If a device's magnetic core exhibits a uniform and constant flux density distribution, B(r), throughout its circumferential magnetic path length, le, as shown by any of its flux density distribution curves then, by the inverse of Ampere's Law's, the magnetic core is constructed with a constant radius of curvature.
For non-circular magnetic core construction geometries, such as a square core, magnetic flux density distributions must conform to Ampere's Law at all points along the core's magnetic path length, le. However, the non-circular shape of the core forces the core's magnetic flux lines to traverse long straight magnetic sections with effectively large radius of curvature (rIDes) and traverse corners with effectively much smaller radius of curvature (rIDec). (rIDes>>rIDec) The square core's straight sections are the dominant regions that determine the non circular device's toroidal shape equivalent effective radius of inner diameter, rIDe, and equivalent effective radius of outer diameter, rODe, respectively, by the inner magnetic path length periphery (lei), the outer magnetic path length periphery (leo), and the requirement that the equivalent toroidal physical size and inductance coincide with the square core's physical size and inductance.
The operational description of redistributed magnetic flux density in a magnetic core assumes that the magnetic material used in the core's cross-section from rIDe, to rODe and at any point along its magnetic path length, le, is ideal and has a constant, uniform, and isotropic relative magnetic permeability, μR, which is greater than 100. Flux density distribution curves shown herein are only a function of the core's geometry and ideal operating frequency, f.
The magnetic core's voltage limitation, VMx(f), is effected by the magnetic flux distribution within the magnetic core. Increased magnetic flux utilization within a given magnetic core material, without magnetic saturation, achieves a higher operating voltage by Faraday's Law and, therefore, higher power density (PD—Watts/volume). Presently, all the magnetic cores in conventional E-M and PM devices are designed to use simple, Amperian, radial (r), magnetic, flux density distribution, B(r), within the core. Consequently, depending on core geometry, from 10% to up to 50% or more of the core is under utilized.
The power electrical transformer was first patented by Gaulard & Gibbs in 1882 and then practically refined by William Stanley in 1886. Since then, optimizing the magnetic design of conventional coil and core electro-magnetic devices, such as transformers, inductors, delay lines, relays, solenoids, motors, and generators, has been limited to conventional electro-magnetic design techniques. Likewise, permanent magnetic device design has followed the trends set by E-M device design. Little has changed in the design of conventional E-M devices other than the introduction of better performing materials and algorithms to speed up the design process.
It is to be understood that both the foregoing general description and the following detailed description are not limiting but are intended to provide further explanation of the novelty claimed. The accompanying drawings, which are incorporated in and constitute part of this specification, are included to illustrate and provide a further understanding of the method and system described herein. Together with the description, the drawings serve to explain the principles of construction and operation.